Max‐type system of difference equations with positive two‐periodic sequences

This paper deals with the form and the periodicity of the solutions of the max‐type system of difference equations where , and are positive two‐periodic sequences and initial values x₀, x _{? 1}, y₀, y _{? 1} ∈ (0, + ∞ ). Copyright © 2014 John Wiley & Sons, Ltd.     

Transition matrices for quantum waveguides with impurities

We consider the electron propagation in a cylindrical quantum waveguide where D is a bounded domain in described by the Dirichlet problem for the Schrödinger operator where x=(x₁, x₂), , is the transversal confinement potential, and is the impurity potential.  We construct the left and right transition matrices and give an numerical algorithm for their calculations based on the spectral parameter power series method.     

Extending LSQR methods to solve the generalized Sylvester‐transpose and periodic Sylvester matrix equations

It is well known that the least‐squares QR‐factorization (LSQR) algorithm is a powerful method for solving linear systems Ax = b and unconstrained least‐squares problem min_x | | Ax ? b | | . In the paper, the LSQR approach is developed to obtain iterative algorithms for solving the generalized Sylvester‐transpose matrix equation the minimum Frobenius norm residual problem and the periodic Sylvester matrix equation Numerical results are given to illustrate the effect of the proposed algorithms. Copyright © 2013 John Wiley & Sons, Ltd.     

Existence and concentration of bound states for a Kirchhoff type problem with potentials vanishing or unbounded at infinity

In this paper, we study the existence and concentration behavior of positive solutions for the following Kirchhoff type equation: where ɛ is a positive parameter, a and b are positive constants, and 3<p<5. Let denotes the ground energy function associated with , , where is regard as a parameter. Suppose that the potential V(x) decays to zero at infinity like |x|^−α with 0<α≤2, we prove the existence of positive solutions u_ɛ belonging to for vanishing or unbounded K(x) when ɛ > 0 small. Furthermore, we show that the solution u_ɛ concentrates at the minimum points of as ɛ→0⁺.     

Global existence and boundedness of classical solutions for a chemotaxis model with consumption of chemoattractant and logistic source

This paper deals with the following chemotaxis system: under homogeneous Neumann boundary conditions in a bounded domain with smooth boundary. Here, δ and χ are some positive constants and f is a smooth function that satisfies with some constants a ?0,b  > 0, and γ  > 1. We prove that the classical solutions to the preceding system are global and bounded provided that Copyright © 2016 John Wiley & Sons, Ltd.     

Forced oscillation of delay difference equations via nonprincipal solution

In this paper, we obtain a new oscillation result for delay difference equations of the form under the assumption that corresponding homogenous equation is nonoscillatory, where τ_n≤n+1. It is observed that the oscillation behaviormay be altered due to presence of the delay. Extensions to forced Emden‐Fowler–type delay difference equations in the sublinear (0<α<1) and the superlinear (1<α) cases are also discussed.     

On the fractional Schrödinger problem with non‐symmetric potential

We study the semilinear equation where 0 < s < 1, , V(x) is a sufficiently smooth non‐symmetric potential with , and ? > 0 is a small number. Letting U be the radial ground state of (?Δ)^sU + U ? U^p=0 in , we build solutions of the form for points ?_j,j = 1,?,m, using a Lyapunov–Schmidt variational reduction. Copyright © 2014 John Wiley & Sons, Ltd.     

Real‐variable characterizations of Musielak–Orlicz–Hardy spaces associated with Schrödinger operators on domains

Let n≥3, Ω be a strongly Lipschitz domain of and L_Ω:=?Δ+V a Schrödinger operator on L²(Ω) with the Dirichlet boundary condition, where Δ is the Laplace operator and the nonnegative potential V belongs to the reverse Hölder class for some q₀>n/2. Assume that the growth function satisfies that ?(x,·) is an Orlicz function, (the class of uniformly Muckenhoupt weights) and its uniformly critical lower type index , where and μ₀∈(0,1] denotes the critical regularity index of the heat kernels of the Laplace operator Δ on Ω. In this article, the authors first show that the heat kernels of L_Ω satisfy the Gaussian upper bound estimates and the Hölder continuity. The authors then introduce the ‘geometrical’ Musielak–Orlicz–Hardy space via , the Hardy space associated with on , and establish its several equivalent characterizations, respectively, in terms of the non‐tangential or the vertical maximal functions or the Lusin area functions associated with L_Ω. All the results essentially improve the known results even on Hardy spaces with p∈(n/(n + δ),1] (in this case, ?(x,t):=t^p for all x∈Ω and t∈[0,∞)). Copyright © 2016 John Wiley & Sons, Ltd.     

Boundedness in a higher‐dimensional chemotaxis system with porous medium diffusion and general sensitivity

This paper deals with the following chemotaxis system: in a bounded domain with smooth boundary under no‐flux boundary conditions, where satisfies for all with l ?2 and some nondecreasing function on [0,∞ ). Here, f (v )∈C ¹([0,∞ )) is nonnegative for all v ?0. It is proved that when , the system possesses at least one global bounded weak solution for any sufficiently smooth nonnegative initial data. This extends a recent result by Wang (Math. Methods Appl. Sci. 2016 39 : 1159–1175) which shows global existence and boundedness of weak solutions under the condition . Copyright © 2017 John Wiley & Sons, Ltd.     

Global well‐posedness and scattering of a generalized nonlinear fourth‐order wave equation

In this paper, we study the global well‐posedness and scattering theory of the solution to the Cauchy problem of a generalized fourth‐order wave equation where if d ?4, and if d ?5. The main strategy we use in this paper is concentration‐compactness argument, which was first introduced by Kenig and Merle to handle the scattering problem vector so as to control the momentum. Copyright © 2017 John Wiley & Sons, Ltd.     

Existence and uniqueness of solutions of sequential nonlinear fractional difference equations with three‐point fractional sum boundary conditions

In this paper, we consider a discrete fractional boundary value problem of the form: where 0 < α,β≤1, 1 < α + β≤2, λ and ρ are constants, γ > 0, , is a continuous function, and E_βx(t) = x(t + β ? 1). The existence and uniqueness of solutions are proved by using Banach's fixed point theorem. An illustrative example is also presented. Copyright © 2014 John Wiley & Sons, Ltd.     

Boundary value problem for p−Laplacian Caputo fractional difference equations with fractional sum boundary conditions

In this paper, we consider a discrete fractional boundary value problem of the form where 0 < α,β≤1, 1 < α + β≤2, 0 < γ≤1, , ρ is a constant, and denote the Caputo fractional differences of order α and β, respectively, is a continuous function, and ?_p is the p‐Laplacian operator. The existence of at least one solution is proved by using Banach fixed point theorem and Schaefer's fixed point theorem. Some illustrative examples are also presented. Copyright © 2015 John Wiley & Sons, Ltd.     

Well‐posedness and unique continuation property for the generalized Ostrovsky equation with low regularity

In this paper, we investigate the initial value problem (IVP henceforth) associated with the generalized Ostrovsky equation as follows: with initial data in the modified Sobolev space . Using Fourier restriction norm method, Tao's [k,Z]?multiplier method and the contraction mapping principle, we show that the local well‐posedness is established for the initial data with (k = 2) and is established for the initial data with (k = 3). Using these results and conservation laws, we also prove that the IVP is globally well‐posed for the initial data with s = 0(k = 2,3). Finally, using complex variables technique and Paley–Wiener theorem, we prove the unique continuation property for the IVP benefited from the ideas of Zhang ZY. et al., On the unique continuation property for the modified Kawahara equation, Advances in Mathematics (China), http://advmath.pku.edu.cn/CN/10.11845/sxjz.2014078b . Copyright © 2015 John Wiley & Sons, Ltd.     

Poisson semigroup characterization and the trace theorem of a class of fractional mean oscillation spaces

In this paper, we use regular wavelets to study the Poisson extension of the fractional mean oscillation spaces . Via a distributional trace operator π_φ, we establish a relation between and a class of harmonic function spaces .     

Time‐harmonic and asymptotically linear Maxwell equations in anisotropic media

This paper is focused on following time‐harmonic Maxwell equation: where is a bounded Lipschitz domain, is the exterior normal, and ω is the frequency. The boundary condition holds when Ω is surrounded by a perfect conductor. Assuming that f is asymptotically linear as , we study the above equation by improving the generalized Nehari manifold method. For an anisotropic material with magnetic permeability tensor and permittivity tensor , ground state solutions are established in this paper. Applying the principle of symmetric criticality, we find 2 types of solutions with cylindrical symmetries in particular for the uniaxial material.     

Gradient estimate for solutions of nonlinear singular elliptic equations below the duality exponent

In this paper, we study the homogeneous Dirichlet problem for an elliptic equation whose simplest model is where , N≥3 is an open bounded set, θ∈]0,1[, and f belongs to a suitable Morrey space. We will show that the Morrey property of the datum is transmitted to the gradient of a solution.     

Forced oscillation of sublinear impulsive differential equations via nonprincipal solution

In this paper, we give new oscillation criteria for forced sublinear impulsive differential equations of the form where γ∈(0,1), under the assumption that associated homogenous linear equation is nonoscillatory.     

A class of vector‐valued dilation‐and‐modulation frames on the half real line

Vector‐valued frames were first introduced under the name of superframes by Balan in the context of signal multiplexing and by Han and Larson from the mathematical aspect. Since then, the wavelet and Gabor frames in have interested many mathematicians. The space models vector‐valued causal signal spaces because of the time variable being nonnegative. But it admits no nontrivial shift‐invariant system and thus no wavelet or Gabor frame since is not a group by addition (not as ). Observing that is a group by multiplication, we, in this paper, introduce a class of multiplication‐based dilation‐and‐modulation ( ) systems, and investigate the theory of frames in . Since is not closed under the Fourier transform, the Fourier transform does not fit . We introduce the notion of Θ_a transform in , and using Θ_a‐transform matrix method, we characterize frames, Riesz bases, and dual frames in and obtain an explicit expression of duals for an arbitrary given frame. An example theorem is also presented.     

Regular attractor for damped KdV‐Burgers equations on R

We study the well‐posedness and dynamic behavior for the KdV‐Burgers equation with a force on R . We establish L ^p ?L ^q estimates of the evolution , as an application we obtain the local well‐posedness. Then the global well‐posedness follows from a uniform estimate for solutions as t goes to infinity. Next, we prove the asymptotical regularity of solutions in space and by the smoothing effect of . The regularity and the asymptotical compactness in L ² yields the asymptotical compactness in by an interpolation arguement. Finally, we conclude the existence of an globalattractor.     

Rotating periodic solutions for second‐order dynamical systems with singularities of repulsive type

In this paper, we study the following second‐order dynamical system: where c ?0 is a constant, and . When g admits a singularity at zero of repulsive type without the restriction of strong force condition, we apply the coincidence degree theory to prove that the system admits nonplanar collisionless rotating periodic solutions taking the form u (t  + T ) = Q u (t ), with T  > 0 and Q an orthogonal matrix under the assumption of Landesman–Lazer type. Copyright © 2016 John Wiley & Sons, Ltd.